Representing Square Matrices with Constant and Identity Matrix - A Guide

  • Thread starter pjunky
  • Start date
  • Tags
    Matrix
In summary: A square matrix A can only be represented as kI if and only if all the diagonal elements of A are equal to k. In all other cases, it is not possible to represent A as kI for any value of k. This can easily be verified by performing the multiplication kI and comparing it with the original matrix A.
  • #1
pjunky
22
0
Is is possible to represent a "square matrix" with the product of the constant and identity matrix of same order of given matrix


EX:-A=[ ] 4x4 matrix

can I make it something like this A=(K) I 4x4

K= constant
I=identity matrix
 
Mathematics news on Phys.org
  • #2
pjunky said:
Is is possible to represent a "square matrix" with the product of the constant and identity matrix of same order of given matrix


EX:-A=[ ] 4x4 matrix

can I make it something like this A=(K) I 4x4

K= constant
I=identity matrix

It's not clear what you are asking. If I is the identity matrix, then (K)I= K. But what doe you mean by the "constant" matrix? A= (K) I= K only if A is already the "constant" matrix itself.
 
  • #3
No. There are tons of trivial counter examples. Any square matrix with components that are *not* along the diagonal.
 
  • #4
HallsofIvy said:
It's not clear what you are asking. If I is the identity matrix, then (K)I= K. But what doe you mean by the "constant" matrix? A= (K) I= K only if A is already the "constant" matrix itself.

for example
if the matrix A is some thing like this:-
2 0 0
0 2 0
0 0 2 =====> A=(2)I
where K=2
I=identity matrix of order 3



Now what I want to know is if matrix A is
a b c
d e f
g h i =======>A=(k)I

is it possible to shrink A in this form for a square matrix
how can I find what exactly k is??
 
  • #5
pjunky said:
for example
if the matrix A is some thing like this:-
2 0 0
0 2 0
0 0 2 =====> A=(2)I
where K=2
I=identity matrix of order 3



Now what I want to know is if matrix A is
a b c
d e f
g h i =======>A=(k)I

is it possible to shrink A in this form for a square matrix
how can I find what exactly k is??

You can't do what you want if the matrix [tex] A [/tex] does not have a form like these:

[tex]
\begin{bmatrix} 4 & 0\\0 & 4 \end{bmatrix}, \quad \begin{bmatrix} -3 & 0 & 0\\0 & -3 & 0\\0 & 0 & -3 \end{bmatrix}
[/tex]

So, even more directly, if you start with

[tex]
\begin{bmatrix} 4 & 2\\-8 & \pi \end{bmatrix}
[/tex]

you cannot write this as [tex] k I_2 [/tex] no matter how imaginative you are in selecting the number [tex] k [/tex].
 
  • #6
I must add - this is the point both HallsOfIvy and Tac-Tics were making.
 
  • #7
yeah I got the point
@ all people thanks for your help
 
  • #8
so you saying that a nxn matrix can only be written as K * I where k is a constant is if it's diagonal elements are the same. (ie a diagonal matrix where the elements in the diagonal is equal)?
 
  • #9
chota said:
so you saying that a nxn matrix can only be written as K * I where k is a constant is if it's diagonal elements are the same. (ie a diagonal matrix where the elements in the diagonal is equal)?
Have you actually tried this multiplication? The number k times I is exactly a matrix with "k" along the diagaonal and zeros everywhere else. Why are you even asking such a question? It's a lot like asking repeatedly if 1+ 1= 3. DO it and see for yourself!
 
  • #10
chota said:
so you saying that a nxn matrix can only be written as K * I where k is a constant is if it's diagonal elements are the same. (ie a diagonal matrix where the elements in the diagonal is equal)?

This is an indubitable fact of mathematics.
 

What is the Matrix?

The Matrix is a concept introduced in the science fiction movie of the same name, which refers to a simulated reality created by machines to keep humans under control.

What is "Matrix help"?

"Matrix help" refers to the assistance or support provided to individuals who are experiencing issues related to the Matrix, such as glitches or other anomalies.

What types of problems can "Matrix help" address?

"Matrix help" can address a wide range of issues, including but not limited to glitches, errors, system malfunctions, and even attempts by the machines to manipulate or control individuals within the Matrix.

How can one access "Matrix help"?

In the movie, the characters are able to access "Matrix help" through phone calls or by using special programs and codes. In reality, "Matrix help" would not be available as the Matrix itself is a fictional concept.

Is "Matrix help" based on real science?

No, "Matrix help" is not based on real science. The concept of the Matrix is purely fictional and has not been proven or supported by any scientific evidence.

Similar threads

Replies
15
Views
1K
  • Precalculus Mathematics Homework Help
Replies
25
Views
886
  • Precalculus Mathematics Homework Help
Replies
25
Views
806
  • Introductory Physics Homework Help
Replies
4
Views
498
  • Linear and Abstract Algebra
Replies
3
Views
996
Replies
1
Views
724
Replies
9
Views
3K
  • Linear and Abstract Algebra
Replies
2
Views
459
  • Precalculus Mathematics Homework Help
2
Replies
57
Views
2K
Back
Top